Vector Algebra

Introduction to Vectors

**Definition**: A quantity that has both **magnitude** (length) and **direction** is called a **vector**. A **directed line segment** AB (written as AB or **a**) represents a vector with:

**Initial point**: A (starting point)

**Terminal point**: B (ending point)

**Magnitude**: |AB| = distance from A to B (always non-negative)

**Direction**: indicated by the arrow from A to B

**Key Distinction**: Scalar quantities (length, mass, time, speed, temperature) have only magnitude. Vector quantities (displacement, velocity, acceleration, force) have both magnitude and direction.

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Position Vector and Coordinate Representation

**Position Vector**: For a point P(x, y, z) in 3D space with respect to origin O(0, 0, 0), the vector **OP** is called the **position vector** of P, denoted as **r** or **p**.

**Magnitude Formula**:

|**r**| = √(x² + y² + z²)

**Direction Cosines**: For position vector **r** = **OP** of point P(x, y, z):

**Direction angles** α, β, γ are angles made with positive x, y, z axes respectively

**Direction cosines** l, m, n are defined as:

l = cos α = x/r

m = cos β = y/r

n = cos γ = z/r

**Fundamental property**: **l² + m² + n² = 1** (always)

**Direction Ratios**: Numbers proportional to direction cosines, denoted a, b, c (usually the components themselves):

For vector **r** = xi + yj + zk, direction ratios are a:b:c = x:y:z

**Important note**: a² + b² + c² ≠ 1 in general (unlike direction cosines)

Relationship: l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²)

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Types of Vectors

**Zero Vector (Null Vector)**: Vector **0** where initial and terminal points coincide. Magnitude = 0, direction is undefined/arbitrary.

**Unit Vector**: Vector with magnitude exactly 1 unit. For any vector **a** ≠ **0**, unit vector is: **â** = **a**/|**a**|

**Coinitial Vectors**: Two or more vectors sharing the same initial point.

**Collinear Vectors**: Vectors parallel to the same line, regardless of magnitude or direction. Vectors **a** and **b** are collinear if and only if **b** = λ**a** for some non-zero scalar λ.

**Equal Vectors**: Vectors **AB** and **CD** are equal (written **AB** = **CD**) if:

Same magnitude: |**AB**| = |**CD**|

Same direction

Position of initial/terminal points is irrelevant

**Negative of a Vector**: Vector **−a** has same magnitude as **a** but opposite direction. Properties:

**a** + (−**a**) = **0**

|−**a**| = |**a**|

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Addition of Vectors

Triangle Law of Vector Addition

When displacement goes from A → B → C, the net displacement from A → C is represented by:

**AC** = **AB** + **BC**

**Algebraic interpretation**: For vectors **a** and **b**, to find **a** + **b**:

1. Position **a** from point A to B

2. Position **b** from B to C (keeping magnitude and direction unchanged)

3. Resultant **a** + **b** = **AC** (from initial point of **a** to terminal point of **b**)

Parallelogram Law of Vector Addition

If **a** and **b** are represented as adjacent sides of a parallelogram ABCD:

**a** = **AB**, **b** = **AD**

Resultant **a** + **b** = **AC** (diagonal through common point)

**AC** = **AB** + **BC** = **AB** + **AD** (since **BC** = **AD**)

Both laws are equivalent and give same result.

Properties of Vector Addition

**Property 1 - Commutative Law**: **a** + **b** = **b** + **a**

Proof: Using parallelogram ABCD where **AB** = **a**, **AD** = **b**, diagonal **AC** = **a** + **b**

From triangle ABC: **AC** = **AB** + **BC** = **a** + **b**

From triangle ADC: **AC** = **AD** + **DC** = **b** + **a** (since **DC** = **AB**)

Therefore **a** + **b** = **b** + **a**

**Property 2 - Associative Law**: (**a** + **b**) + **c** = **a** + (**b** + **c**)

Proof: Represent vectors as successive displacements

Whether you group them left or right, final position is same

Allows writing **a** + **b** + **c** without parentheses

**Additive Identity**: **a** + **0** = **0** + **a** = **a**

Zero vector is additive identity

**Vector Difference**: **a** − **b** = **a** + (−**b**)

Geometric: If we place **−b** (opposite direction of **b**) at terminal point of **a**, then **a** − **b** is resultant

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Multiplication of Vector by Scalar

**Definition**: For vector **a** and scalar λ, the product λ**a** is a vector such that:

**Magnitude**: |λ**a**| = |λ| · |**a**|

**Direction**:

Same as **a** if λ > 0

Opposite to **a** if λ < 0

Undefined if λ = 0 (gives zero vector)

**Important Cases**:

λ = 0: 0**a** = **0**

λ = 1: 1**a** = **a**

λ = −1: −**a** is negative of **a** (opposite direction, same magnitude)

λ = 1/|**a**|: λ**a** = unit vector **â**

**Collinearity via Scalar Multiplication**: Vectors **a** and **b** are collinear ⟺ **b** = λ**a** for some non-zero λ ∈ ℝ

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Components of a Vector - Standard Unit Vectors

**Unit Vectors Along Axes**:

**î** = unit vector along x-axis = (1,0,0)

**ĵ** = unit vector along y-axis = (0,1,0)

**k̂** = unit vector along z-axis = (0,0,1)

Each has magnitude 1: |**î**| = |**ĵ**| = |**k̂**| = 1

**Component Form of Position Vector**: For point P(x, y, z), position vector is:

**r** = x**î** + y**ĵ** + z**k̂**

**Scalar Components**: x, y, z (the coefficients)

**Vector Components**: x**î**, y**ĵ**, z**k̂** (components along each axis)

**Magnitude from Components**:

|**r**| = √(x² + y² + z²)

Operations in Component Form

For vectors **a** = a₁**î** + a₂**ĵ** + a₃**k̂** and **b** = b₁**î** + b₂**ĵ** + b₃**k̂**:

**Addition**:

**a** + **b** = (a₁ + b₁)**î** + (a₂ + b₂)**ĵ** + (a₃ + b₃)**k̂**

**Subtraction**:

**a** − **b** = (a₁ − b₁)**î** + (a₂ − b₂)**ĵ** + (a₃ − b₃)**k̂**

**Scalar Multiplication**:

λ**a** = λa₁**î** + λa₂**ĵ** + λa₃**k̂**

**Equality**: **a** = **b** ⟺ a₁ = b₁, a₂ = b₂, a₃ = b₃

**Collinearity**: **a** and **b** are collinear ⟺ a₁/b₁ = a₂/b₂ = a₃/b₃ (provided denominators ≠ 0)

Unit Vector in a Given Direction

**Formula**: **â** = **a**/|**a**| = **a**/√(a₁² + a₂² + a₃²)

**Example**: For **a** = 2**î** + 3**ĵ** + **k̂**

|**a**| = √(4 + 9 + 1) = √14

**â** = (2**î** + 3**ĵ** + **k̂**)/√14 = (2/√14)**î** + (3/√14)**ĵ** + (1/√14)**k̂**

Direction Ratios and Direction Cosines from Components

For **a** = a₁**î** + a₂**ĵ** + a₃**k̂**:

**Direction ratios**: a₁ : a₂ : a₃

**Direction cosines**:

l = a₁/√(a₁² + a₂² + a₃²)

m = a₂/√(a₁² + a₂² + a₃²)

n = a₃/√(a₁² + a₂² + a₃²)

**Example**: For **a** = **î** + **ĵ** − 2**k̂**

Direction ratios: 1:1:(−2)

|**a**| = √(1 + 1 + 4) = √6

Direction cosines: l = 1/√6, m = 1/√6, n = −2/√6

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Vector Joining Two Points

**Formula**: For points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), the vector from P₁ to P₂ is:

**P₁P₂** = (x₂ − x₁)**î** + (y₂ − y₁)**ĵ** + (z₂ − z₁)**k̂**

**Derivation**: Using position vectors:

**P₁P₂** = **OP₂** − **OP₁** = **r₂** − **r₁**

**Magnitude**: |**P₁P₂**| = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] (distance formula)

**Example**: Vector from P(2, 3, 0) to Q(−1, −2, −4):

**PQ** = (−1−2)**î** + (−2−3)**ĵ** + (−4−0)**k̂** = −3**î** − 5**ĵ** − 4**k̂**

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Section Formula

Internal Division

**Theorem**: Point R divides line segment PQ **internally** in ratio m:n ⟺ **PR** : **RQ** = m : n

**Position Vector Formula**:

**r** = (m**r₂** + n**r₁**)/(m + n)

where **r₁** and **r₂** are position vectors of P and Q respectively.

**Key insight**: Larger weight (m) is on terminal point (Q), smaller weight (n) on initial point (P).

**Midpoint Formula** (when m = n):

**r** = (**r₁** + **r₂**)/2

External Division

**Theorem**: Point R divides line segment PQ **externally** in ratio m:n ⟺ **PR** : **RQ** = m : n (with R outside segment)

**Position Vector Formula**:

**r** = (m**r₂** − n**r₁**)/(m − n)

**Note**: m > n for external division; if m = n, denominators become zero (points at infinity).

**Example - Internal Division**: P with **r₁** = 2**î** − **ĵ** + **k̂** and Q with **r₂** = **î** + 3**ĵ** − 5**k̂**, find R dividing PQ in ratio 2:1 internally:

**r** = [2(**î** + 3**ĵ** − 5**k̂**) + 1(2**î** − **ĵ** + **k̂**)]/(2+1)

= (4**î** + 5**ĵ** − 9**k̂**)/3

**Example - External Division**: Same P and Q, ratio 2:1 externally:

**r** = [2(**î** + 3**ĵ** − 5**k̂**) − 1(2**î** − **ĵ** + **k̂**)]/(2−1)

= **0î** + 7**ĵ** − 11**k̂**

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Distributive Laws for Scalar Multiplication

For vectors **a**, **b**, **c** and scalars k, m:

**Law 1**: k(**a** + **b**) = k**a** + k**b**

**Law 2**: (k + m)**a** = k**a** + m**a**

**Law 3**: k(m**a**) = (km)**a**

**Proof concept**: These follow from component-wise operations and distributivity of real number multiplication.

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Important Exam Points and Common Mistakes

**Do Not Confuse**:

|−**a**| = |**a**| (magnitudes are always equal)

But −**a** ≠ **a** (different directions)

**Direction Cosines vs Ratios**:

Direction cosines always satisfy l² + m² + n² = 1

Direction ratios do NOT necessarily satisfy this

To convert ratios a, b, c to cosines: divide by √(a² + b² + c²)

**Equality of Vectors**:

Requires all three components to match

Position is irrelevant for free vectors

**a** + **b** = **c** + **d** does NOT imply **a** = **c** or **b** = **d**

**Collinearity Condition**:

**a** || **b** ⟺ **a** = λ**b** for non-zero λ

In component form: a₁/b₁ = a₂/b₂ = a₃/b₃

Works only if denominators are non-zero

**Section Formula Signs**:

Internal: both coefficients positive, sum in denominator

External: coefficients have opposite signs, difference in denominator

Remember: m is weight at terminal point, n at initial